Anisotropic potential of velocity fields in real fluids: Application to the MAST solution of shallow water equations

In the present paper it is first shown that, due to their structure, the general governing equations of uncompressible real fluids can be regarded as an “anisotropic” potential flow problem and closed streamlines cannot occur at any time. For a discretized velocity field, a fast iterative procedure is proposed to order the computational elements at the beginning of each time level, allowing a sequential solution element by element of the advection problem. Some closed circuits could appear due to the discretization error and the elements involved in these circuits could not be ordered. We prove in the paper that the total flux of these not ordered elements goes to zero by refining the computational mesh and that it is possible to order all the remaining elements by neglecting the minimum inter-element flux inside each circuit, with a very small resulting error. The methodology is then applied to the solution of the 2D shallow water equations. The governing Partial Differential Equations are discretized over a generally unstructured triangular mesh, which attains the generalised Delaunay property. Solution is obtained applying a prediction-correction time step procedure. The prediction problem is solved applying a MArching in Space and Time (MAST) procedure, where the computational elements are required to be ordered and explicitly solved. In the correction step, a large linear well-conditioned system is solved. Model results are compared with experimental data and other numerical literature results. Computational costs have been estimated and the convergence order has been investigated according to a known exact solution.